Probability Theory II
نویسنده
چکیده
Contents Chapter 1. Martingales, continued 1 1.1. Martingales indexed by partially ordered sets 1 1.2. Notions of convergence for martingales 3 1.3. Uniform integrability 4 1.4. Convergence of martingales with directed index sets 6 1.5. Application: The 0-1 law of Kolmogorov 8 1.6. Continuous-time martingales 9 1.7. Tail bounds for martingales 12 1.8. Application: The Pólya urn 13 1.9. Application: The Radon-Nikodym theorem 15 Chapter 2. Measures on nice spaces 19 2.1. Topology review 19 2.2. Metric and metrizable spaces 22 2.3. Regularity of measures 23 2.4. Weak convergence 25 2.5. Polish spaces and Borel spaces 28 2.6. The space of probability measures 31 2.7. Compactness and local compactness 34 2.8. Tightness 36 2.9. Some additional useful facts on Polish spaces 38
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